A334094 -id:A334094 - cp's OEIS frontend

A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

1, 2, 3, 4, 5, 7, 8, 6, 9, 19, 16, 10, 11, 21, 43, 32, 12, 13, 23, 47, 127, 64, 17, 14, 27, 49, 129, 283, 128, 20, 15, 29, 57, 133, 301, 659, 256, 24, 18, 31, 59, 139, 329, 817, 1319, 512, 34, 22, 33, 63, 141, 343, 827, 1699, 3957, 1024, 40, 25, 35, 67, 147, 347, 839, 1787, 4079, 9227, 2048, 48, 26, 37, 69, 161, 361, 849, 1849, 4613, 9233, 21599

Offset: 1

Antti Karttunen, Apr 14 2020

Array is read by descending antidiagonals with (n,k) = (0,1), (0,2), (1,1), (0,3), (1,2), (2,1), ... where A(n,k) is the k-th solution x to A329697(x) = n. The row indexing (n) starts from 0, and column indexing (k) from 1.
Any odd prime that appears on row n is 1+{some term on row n-1}.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A329697 is completely additive.
The binary weight (A000120) of any term on row n is at most 2^n.

			The top left corner of the array:
  n\k |    1     2     3     4     5     6     7     8     9    10
------+----------------------------------------------------------------
   0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
   1  |    3,    5,    6,   10,   12,   17,   20,   24,   34,   40, ...
   2  |    7,    9,   11,   13,   14,   15,   18,   22,   25,   26, ...
   3  |   19,   21,   23,   27,   29,   31,   33,   35,   37,   38, ...
   4  |   43,   47,   49,   57,   59,   63,   67,   69,   71,   77, ...
   5  |  127,  129,  133,  139,  141,  147,  161,  163,  171,  173, ...
   6  |  283,  301,  329,  343,  347,  361,  379,  381,  383,  387, ...
   7  |  659,  817,  827,  839,  849,  863,  883,  889,  893,  903, ...
   8  | 1319, 1699, 1787, 1849, 1977, 1979, 1981, 2021, 2039, 2083, ...
   9  | 3957, 4079, 4613, 4903, 5097, 5179, 5361, 5377, 5399, 5419, ...
etc.
Note that the row 9 is the first one which begins with composite, as 3957 = 3*1319. The next such rows are row 15 and row 22. See A334099.

Index entries for sequences that are permutations of the natural numbers

Cf. A329697.
Cf. A334099 (the leftmost column).
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for the rows 0-6.
Cf. A019434, A334092, A334093, A334094, A334095, A334096 for the primes on the rows 1-6.
Cf. also irregular triangle A334111.

Block[{nn = 16, s}, s = Values@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, 2^nn]; Table[s[[#, k]] &[m - k + 1], {m, nn - Ceiling[nn/4]}, {k, m, 1, -1}]] // Flatten (* Michael De Vlieger, Apr 30 2020 *)

up_to = 105; \\ up_to = 1081; \\ = binomial(46+1,2)
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
memoA334100sq = Map();
A334100sq(n, k) = { my(v=0); if(!mapisdefined(memoA334100sq,[n,k-1],&v),if(1==k, v=0, v = A334100sq(n, k-1))); for(i=1+v,oo,if(A329697(i)==(n-1),mapput(memoA334100sq,[n,k],i); return(i))); };
A334100list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A334100sq(col,(a-(col-1))))); (v); };
v334100 = A334100list(up_to);
A334100(n) = v334100[n];

A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

Original entry on oeis.org

7, 11, 13, 41, 97, 137, 193, 641, 769, 12289, 40961, 163841, 557057, 786433, 167772161, 2281701377, 3221225473, 206158430209, 2748779069441, 6597069766657, 38280596832649217, 180143985094819841, 221360928884514619393, 188894659314785808547841, 193428131138340667952988161

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p such that p-1 is not a power of two, but for which A171462(p-1) = (p-1-A052126(p-1)) is [a power of 2].

Primes of the form ((2^(2^k))+1)*2^h + 1, where ((2^(2^k))+1) is one of the Fermat primes, A019434, 3, 5, 17, 257, ..., .

Charles R Greathouse IV, Table of n, a(n) for n = 1..53

Primes in A334102.
Intersection of A081091 and A147545.
Subsequences: A039687, A050526, A300407.
Cf. A000040, A010051, A019434, A052126, A171462, A329697. Cf. also A334093, A334094, A334095, A334096.

isA334092(n) = (isprime(n)&&2==A329697(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334092(n) = (isprime(n)&&(!A209229(n-1))&&A209229(n-1-A052126(n-1)));

list(lim)=if(exponent(lim\=1)>=2^33, error("Verify composite character of more Fermat primes before checking this high")); my(v=List(),t); for(e=0,4, t=2^2^e+1; while((t<<=1)Charles R Greathouse IV, Apr 14 2020

More terms from Giovanni Resta, Apr 14 2020

A334103 Numbers n for which A329697(n) == 3.

Original entry on oeis.org

19, 21, 23, 27, 29, 31, 33, 35, 37, 38, 39, 42, 45, 46, 53, 54, 55, 58, 61, 62, 65, 66, 70, 73, 74, 75, 76, 78, 83, 84, 89, 90, 92, 101, 103, 106, 108, 110, 113, 116, 119, 122, 123, 124, 125, 130, 132, 140, 146, 148, 150, 152, 153, 156, 166, 168, 178, 180, 184, 187, 202, 205, 206, 212, 216, 220, 221, 226, 232, 238, 241, 244

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers n for which A171462(n) = n-A052126(n) is in A334102.

Among the first 2821 terms (terms < 2^31), there are terms with binary weights 2, 3, 4, 5, 6 and 8. For example, 33 is the first term with binary weight 2, and 255 is the first term with binary weight 8.

Antti Karttunen, Table of n, a(n) for n = 1..2821; all terms <= 2^31

Row 3 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334104, A334106.
Cf. A334093 (primes present), A334094.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334103(n) = (3==A329697(n));

A334104 Numbers m for which A329697(m) = 4.

Original entry on oeis.org

43, 47, 49, 57, 59, 63, 67, 69, 71, 77, 79, 81, 86, 87, 91, 93, 94, 95, 98, 99, 105, 107, 109, 111, 114, 115, 117, 118, 121, 126, 131, 134, 135, 138, 142, 143, 145, 149, 151, 154, 155, 157, 158, 159, 162, 165, 167, 169, 172, 174, 175, 179, 181, 182, 183, 185, 186, 188, 190, 195, 196, 198, 210, 214, 218, 219, 222, 225

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Squares of A334102 form a subsequence.

Among the first 12193 terms (terms < 2^31), there are terms with binary weights 2 - 16, except no terms with weight 13, 14 or 15. For example, 1025 is the first term with binary weight 2, and 65535 is the first term with binary weight 16.

			63 = 7*9 is a term as both 7 and 9 are terms of A334102.
65535 = 3*5*17*257 is a term as it is a product of four Fermat primes, thus in four steps all odd primes can be eliminated with p -> (p-1) map.

Antti Karttunen, Table of n, a(n) for n = 1..12193; all terms <= 2^31

Row 4 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334105, A334106.
Cf. A334094 (primes present).

Position[Array[Length@NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 225], 4][[All, 1]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334104(n) = (4==A329697(n));

A334093 Primes p for which A329697(p) == 3.

Original entry on oeis.org

19, 23, 29, 31, 37, 53, 61, 73, 83, 89, 101, 103, 113, 241, 353, 389, 401, 409, 449, 577, 773, 1097, 1153, 1283, 1361, 1409, 1543, 1553, 1601, 3089, 3329, 5441, 6529, 7681, 13313, 15361, 17477, 18433, 25601, 26113, 49157, 49409, 61441, 82241, 83969, 87041, 98689, 114689, 147457, 295937, 327689, 328961, 417793

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334102(n) + 1, for some n >= 1.

Cf. A329697, A334102, primes in A334103.

Cf. also A019434, A334092, A334094, A334095, A334096.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334093(n) = (isprime(n)&&(3==A329697(n)));

A334095 Primes p for which A329697(p) == 5.

Original entry on oeis.org

127, 139, 163, 173, 191, 197, 199, 211, 223, 229, 263, 269, 271, 277, 311, 317, 331, 349, 359, 367, 373, 397, 421, 439, 457, 461, 467, 479, 499, 503, 541, 563, 569, 587, 607, 617, 619, 647, 661, 677, 701, 733, 739, 751, 761, 857, 877, 887, 919, 937, 997, 1009, 1031, 1049, 1061, 1069, 1123, 1187, 1193, 1213, 1229, 1231

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334104(n) + 1, for some n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..1454; all terms < 2^31

Primes in A334105.
Cf. A000040, A010051, A052126, A171462, A329697, A334104.
Cf. also A019434, A334092, A334093, A334094, A334096.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334095(n) = (isprime(n)&&(5==A329697(n)));

A334096 Primes p for which A329697(p) == 6.

Original entry on oeis.org

283, 347, 379, 383, 419, 431, 463, 487, 491, 509, 523, 547, 557, 571, 599, 631, 643, 653, 683, 691, 709, 719, 727, 743, 757, 787, 797, 811, 829, 853, 859, 907, 911, 941, 991, 1013, 1033, 1051, 1087, 1091, 1093, 1109, 1117, 1129, 1151, 1163, 1171, 1181, 1277, 1289, 1381, 1399, 1451, 1453, 1493, 1511, 1523, 1559, 1571, 1583, 1607

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334105(n) + 1, for some n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..4827; all terms < 2^31

Primes in A334106.
Cf. A000040, A010051, A052126, A171462, A329697, A334105.
Cf. also A019434, A334092, A334093, A334094, A334095.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334096(n) = (isprime(n)&&(6==A329697(n)));

A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 167, 179, 193, 227, 233, 257, 353, 359, 389, 449, 467, 641, 719, 769, 773, 857, 929, 1097, 1283, 1409, 1433, 1439, 1553, 1697, 1889, 2657, 2819, 2879, 3089, 3329, 3593, 3617, 3779, 5639

Offset: 1

T. D. Noe, Nov 07 2008

nonn

This sequence starts like A074781 but grows much faster. Observe that there can be large differences between consecutive terms. Can it be shown that there is always such a prime between consecutive powers of 2? Or that this sequence is infinite? By theorem 1 of the Noe paper, this sequence is a subsequence of A135832, primes in Section I of the phi iteration.
From Antti Karttunen, Apr 19 2020: (Start)
Sequence can be considered as a generalization of Fermat primes, A019434, which is a subsequence of this sequence.
All terms with binary weight k (A000120, at least 2 for these terms) can be found as a subset of primes found on the row k-1 of array A334100. E.g. primes with weight 2 are Fermat primes (A019434), those with weight 3 are A334092 (which doesn't contain any other primes), those with weight 4 are in A334093 (among also other kind of primes), those with weights 5, 6, 7 are included as (proper) subsets in A334094, A334095 and A334096 respectively. (End)

T. D. Noe, Table of n, a(n) for n=1..2000
T. D. Noe, Primes in classes of the iterated totient function, J. Integer Sequences, 11 (2008), Article 08.1.2.

Cf. A000265, A329697, A334100.
Subsequence of A074781, and of A135832.
Subsequences: A019434, A334092 (including A039687, A050526, A300407).

nn=2^13; t={1}; i=1; While[q=t[[i]]; k=1; While[p=1+q*2^k; p

A000265(n) = (n>>valuation(n,2));
isA147454(n) = ((n>2)&&isprime(n)&&((1==(n=A000265(n-1)))||isA147454(n))); \\ Antti Karttunen, Apr 19 2020

A329697(a(n)) = A000120(a(n)) - 1. - Antti Karttunen, Apr 19 2020

cp's OEIS Frontend

A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

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A334092 Primes p of the form of the form q*2^h + 1, where q is one of the Fermat primes; Primes p for which A329697(p) == 2.

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A334103 Numbers n for which A329697(n) == 3.

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A334104 Numbers m for which A329697(m) = 4.

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A334093 Primes p for which A329697(p) == 3.

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A334095 Primes p for which A329697(p) == 5.

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A334096 Primes p for which A329697(p) == 6.

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A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

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